Calculate The Double Integral Xsin X Y

This guide explains how to calculate the double integral of xsin(x)y, including the formula, step-by-step instructions, and practical applications. Use the interactive calculator in the sidebar to compute specific values quickly.

What is a Double Integral?

A double integral extends the concept of single integration to two dimensions. It calculates the volume under a surface defined by a function over a region in the xy-plane. For the function xsin(x)y, we'll integrate with respect to y first, then x.

Double integrals are used in physics, engineering, and economics to calculate quantities like mass, charge, or average values over a region.

Formula for xsin(x)y

The double integral of xsin(x)y over a region R is calculated as:

∫∫_R x sin(x) y dA = ∫_a^b [∫_c^{d x sin(x) y dy] dx}

Where:

x and y are the variables of integration
R is the region of integration
dA represents the infinitesimal area element

How to Calculate

Identify the region of integration R
Set up the iterated integral with respect to y first, then x
Integrate the inner integral with respect to y
Integrate the result with respect to x
Evaluate the definite integral using the limits of integration

For complex regions, you may need to use polar coordinates or other coordinate transformations.

Worked Example

Let's calculate the double integral of xsin(x)y over the region where 0 ≤ x ≤ π and 0 ≤ y ≤ sin(x).

∫₀^π [∫₀^sin(x) x sin(x) y dy] dx

First, integrate with respect to y:

∫₀^sin(x) x sin(x) y dy = x sin(x) [y²/2]₀^sin(x) = x sin(x) (sin²(x)/2)

Then integrate with respect to x:

∫₀^π (x sin(x) sin²(x)/2) dx = (1/2) ∫₀^π x sin³(x) dx

The final result is approximately 0.089.

Applications

The double integral of xsin(x)y has applications in:

Physics for calculating work done by variable forces
Engineering for finding centers of mass
Economics for calculating average values over regions

FAQ

What is the difference between single and double integrals?

Single integrals calculate area under a curve, while double integrals calculate volume under a surface over a region in two dimensions.

When would I use a double integral instead of a single integral?

Use double integrals when you need to calculate quantities over two-dimensional regions, such as volume, mass, or average values.

How do I handle complex regions of integration?

For complex regions, you may need to use coordinate transformations, such as polar coordinates, to simplify the integral.